Panel Regression Model

Dan Ovando

October 21, 2015

The Panel Regression Model

Panel regression:
- Fancy term for regression with time
- Also called “longitudinal” data
- Track objects of interest over time
- E.g. Costello et al. 2008: Do catch shares work?

\[log(\frac{B}{B_{msy}})_{i,y} = \alpha + \beta{X_{i}} + \gamma{Y_{i,y}} + \epsilon_{s} + e_{i,y,s}\]

Basic linear regression fit via OLS
- Most other regression methods are basically trying correct the SEs
Uses
- Time
- Catch
- Life history

**Model 1**

	Dependent variable:

	LogBvBmsy

YearsBack	0.01^***
	(0.001)

ScaledCatch	0.29^*
	(0.16)

ScaledCatch1Back	0.20
	(0.17)

ScaledCatch2Back	0.07
	(0.17)

ScaledCatch3Back	0.04
	(0.16)

ScaledCatch4Back	-0.12
	(0.13)

MaxCatch	0.0000^*
	(0.0000)

TimeToMaxCatch	0.02^***
	(0.001)

InitialScaledCatchSlope	-4.05^***
	(0.41)

MeanScaledCatch	0.63^***
	(0.16)

CatchToRollingMax	0.91^***
	(0.10)

MaxLength	0.0005^**
	(0.0002)

AgeMat	0.03^***
	(0.005)

VonBertK	-0.18^*
	(0.10)

Temp	0.03^***
	(0.01)

SpeciesCatNameFlounders, halibuts, soles	0.72^***
	(0.07)

SpeciesCatNameHerrings, sardines, anchovies	0.61^***
	(0.06)

SpeciesCatNameMiscellaneous coastal fishes	0.67^***
	(0.20)

SpeciesCatNameMiscellaneous demersal fishes	1.17^***
	(0.10)

SpeciesCatNameMiscellaneous pelagic fishes	-0.42^***
	(0.10)

SpeciesCatNameSharks, rays, chimaeras	0.80^***
	(0.23)

SpeciesCatNameTunas, bonitos, billfishes	0.44^***
	(0.09)

Constant	-2.75^***
	(0.11)


Observations	3,240
R²	0.48
Adjusted R²	0.47
Residual Std. Error	0.86 (df = 3217)
F Statistic	132.43^*** (df = 22; 3217)

Note:	p<0.1; p<0.05; p<0.01

In log space

Now exponentiated

Residuals vs. predicted

Residuals vs. exponentiated predicted values

Residuals over time

We have a few problems

Means and medians are biased
Errors…
- are clusterted
- heteroskedasticity
Solutions
- for the mean, correct by smearing estimate \(\sigma^2/2\)
- for the median…

denote \(log(B/B_{MSY}) = x_{i,t}\beta + e_{i,t}\) ~ \(N(0,\sigma_{i,t}^{2})\)
- allows for heterskedasticity
Simulate an N X J matrix where N is the number of X’s and J is the # of bootstraps
- Estimate at [N,J] is \(\sigma_{i,t} = x_{i,t}\beta + e_{i,t}\) ~ \(N(0,\phi_{i,t}^{2})\)
- \(\phi\) is time and B/B_msy specific
- Take the median of each column
- Exponentiate to get an unbiased distribution around the median